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An almost-sure estimate for the mean of generalized Q -multiplicative functions of modulus 1

Jean-Loup Mauclaire — 2000

Journal de théorie des nombres de Bordeaux

Let Q = ( Q k ) k 0 , Q 0 = 1 , Q k + 1 = q k Q k , q k 2 , be a Cantor scale, 𝐙 Q the compact projective limit group of the groups 𝐙 / Q k 𝐙 , identified to 0 j k - 1 𝐙 / q j 𝐙 , and let μ be its normalized Haar measure. To an element x = { a 0 , a 1 , a 2 , } , 0 a k q k + 1 - 1 , of 𝐙 Q we associate the sequence of integral valued random variables x k = 0 j k a j Q j . The main result of this article is that, given a complex 𝐐 -multiplicative function g of modulus 1 , we have lim x k x ( 1 x k n x k - 1 g ( n ) - 0 j k 1 q j 0 a q j g ( a Q j ) ) = 0 μ -a.e .

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